Growth property of singular solutions of linear partial differential equations in the complex domain in $C^{d+1}$ (Complex Analysis and Microlocal Analysis)

大内 忠 (上智大)

Growth property of singular solutions of linear partial

differential equations in the complex domain in $C^{d+1}$

By

Sunao

$\overline{\mathrm{o}}_{\mathrm{U}\mathrm{c}}\mathrm{H}\mathrm{I}$

(Sophia Univ.)

ABSTRACT. Let $P(z, \partial)$ be a linear partial differential operator

with coefficients holomophic in a neighbourhood $\Omega$ of $z=0$ in

$C^{d+1}$. Consider equation _{$P(z, \partial)u(z)=f(z)$}, where_{$u(z)$ and $f(z)$}

admit singularities on the surface $\{z_{0}=0\}$. We assume that

$|f(z)|\leq A|z_{0}|^{C}$ in a region $\Omega(\theta)$ which is sectorial with respect

to$z_{0}$. The main result of this paper is the following:

There is an exponent $\gamma^{*}$ such that for some class of operators

if $\forall\epsilon>0\exists C_{\epsilon}$ such that $|u(Z)|\leq c_{\epsilon}\exp(\epsilon|Z0|^{-\gamma^{*}})$ in $\Omega(\theta)$, then $|u(Z)|\leq C|z_{0}|^{\mathrm{C}^{;}}$ for some constants $c’$ and $C$.

First

we

give the notations briefly. The coordinates of $C^{d+1}$ are

de-noted by $z=$ $(z_{0,1}z, \cdots , z_{d})=(Z_{0}, Z’)\in C\cross C^{d}$. $|z|= \max\{|Z_{i}|;0\leq$

$i\leq d\}$ and $|z’|= \max\{|Z_{i}|; 1 \leq i\leq d\}$. Its dual variables are

$\xi=(\xi_{0}, \xi’)=(\xi_{0}, \xi_{1}, \cdots, \xi_{d})$

.

$N$ is the set of all nongegative

inte-gers $N=\{0,1,2, \cdots\}$. The differentiation is denoted by $\partial_{i}=\partial/\partial z_{i}$,

and $\partial=$ $(\partial_{0}, \partial_{1}, \cdots , \partial_{d})=(\partial_{0}, \partial’)$. For a multi-indes $\alpha=(\alpha_{0}, \alpha’)\in$

$N\cross N^{d},$ $| \alpha|=\alpha_{0}+|\alpha’|=\sum_{i=0^{\alpha}}^{d}i$. Define $\partial^{\alpha}=\prod_{i=0}^{d}\partial^{\alpha}ii$. We denote $\partial^{J\alpha’}=\prod_{i=1}^{d}\partial^{\alpha_{i}}i$ by $\partial^{\alpha’}$

.

We define spaces of holomorphic functions in some regions to state

the results. Let $\Omega=\Omega_{0}\cross\Omega’$ be

a

polydisk with $\Omega_{0}=\{z_{0}\in C^{1}$; _{$|z_{0}|<$} $R\}$ and $\Omega’=\{z’\in C^{d};|z’|<R\}$ for

some

positive constant $R$. Put

$\Omega_{0}(\theta)=\{z_{0}\in\Omega_{0}-\{0\}; |\arg_{Z_{0}}|<\theta\}$ and $\Omega(\theta)=\Omega_{0}(\theta)\cross\Omega’$.

$\mathcal{O}(\Omega)(\mathcal{O}(\Omega’), \mathcal{O}(\Omega(\theta)))$ is the set of all holomorphic functions on

$\Omega$ (resp. $\Omega’,$ $\Omega(\theta)$) . $\mathcal{O}(\Omega(\theta))$ contains multi-valued functions, if $\theta>\pi$.

We introduce $\mathcal{O}_{(\kappa)}(\Omega(\theta))$ and $Asy_{\{\kappa\}}(\Omega(\theta))$, which

are

subspaces of

$O(\Omega(\theta))$ and fundamental function spaces in this paper.

Definition 1. $\mathcal{O}_{(\kappa)}(\Omega(\theta))(0<\kappa<+\infty)$ is the set

of

all _$u(z)\in$ $\mathcal{O}(\Omega(\theta))$ such that

for

any $\epsilon>0$ and any $\theta’$ with $0<\theta’<\theta$

(1) $|u(_{Z)|}\leq C\exp(\epsilon|z_{0}|^{-}\kappa)$ $z\in\Omega(\theta/)$

holds

_for

a constant $C=C(\epsilon, \theta’)$. We put $O_{(+\infty)}(\Omega(\theta))=\mathcal{O}(\Omega(\theta))$

for

$\kappa=+\infty$.

Definition 2. $\mathcal{O}_{reg,c}(\Omega(\theta))$ is the set

of

all

$u(z.)\in O(\Omega(\theta))$ such

that any $\theta’$ with _{$0<\theta’<\theta$}

(2) $|u(_{Z)|}\leq C|_{Z}0|^{c}$ $z\in\Omega(\theta/)$

holds

_for

a

constant $C=C(\theta’)$.

We say that $u(z)\in O(\Omega(\theta))$ is slowly increasing in $\Omega(\theta)$, if$u(z)\in$

$\cup$ _{$O_{reg,c}(\Omega(\theta))$}.

$|c|<+\infty$

数理解析研究所講究録

Now let $P(z, \partial)$ be an m-th order linear partial differential equation

with coeficients in $O(\Omega)$

(3) $P(z, \partial)=\sum a\alpha(z)\partial\alpha=\sum_{m|\alpha|\leq m|\alpha|\leq}z_{0^{\alpha}}bj(\alpha z)\partial^{\alpha}$,

where $j_{\alpha}\in N$ is the valuation of $a_{\alpha}(z)$ with respect to $z_{0},$ $a_{\alpha}(z)=$

$z_{0}^{j\alpha}b_{\alpha}(z)$. Let

us

define

some

quantities for $P(z, \partial)$:

(4)

$\{$

$e_{*}:= \min\{j_{\alpha}-\alpha_{0;}\alpha\in Nd+1\},$ $\triangle=\{\alpha\in N^{d+}1j_{\alpha}-\alpha_{0=}e*\}$; $k^{*}:= \max\{|\alpha|;\alpha\in\triangle\}$.

Put

(5) $\mathfrak{P}(z, \partial)=\sum_{\alpha\in\triangle}z_{00(Z’}^{j\alpha}b,)\partial^{\alpha}\mathrm{Q}$.

Let

us

introduce

an

index which plays an important role in this paper.

Definition 3. (Minimal irregularity)

(6) $\{_{\gamma^{*}}^{\gamma^{*}}.\cdot.\cdot=\min_{=\infty},$$\{\frac{j_{\alpha}-\alpha_{0^{-e_{*}}}}{ifk=m|\alpha|-k^{*}} ; \alpha*.\in Nd+1, |\alpha|>k^{*}\}$

,

_if

$k^{*}<m$,

Let us introduce conditions

on

$P(z, \partial)$.

Condition $0$

.

If

$\alpha=(\alpha_{0}, \alpha)’\in\triangle$, then $\alpha’=(0,0, \cdots , 0)$. The following condition is

more

strict than Condition $0$

.

Condition 1. $P(z, \partial)$

satisfies

Condition $\mathit{0}$ and

$b_{(k^{\mathrm{x}},0},0,\cdots,0$₎(0) $\neq 0$

.

Suppose that $P(z, \partial)$ satisfies Condition $0$. Then $\mathfrak{P}(z, \partial)$ is an

ordi-nary differential operator,

(7) $\mathfrak{P}(z, \partial)=\sum_{\alpha\in\triangle}Z_{0^{*}\alpha,0(Z’}^{e}b)_{Z^{\alpha 0}}0\partial_{0}\alpha_{0}$ ,

and $\{z_{0}=0\}$ isregular singular. Define the indicial polynomial$\chi_{P}(z’, \lambda)$

$\mathrm{o}\mathrm{f}\mathfrak{P}(Z, \partial)$,

(8) $\chi_{P}(_{Z’}, \lambda):=\sum b_{\alpha,0}(z)\lambda(\lambda-1)\cdots(\lambda-\alpha_{0}+1)\alpha\in\triangle/$ .

Further suppose that $P(z, \partial)$ satisfies Condition 1. Then $\chi_{P}(Z’, \lambda)$ is

a

polynomial of $\lambda$ with degree $k^{*}$ in $\{z;|z|\leq R\}$. Hence there exist

real constants $a_{0},$$a_{1}$ and $b_{0}$ such that all the roots of$\chi_{P}(z’,$ $\lambda \mathrm{I}=0$ for

$|z|\leq R$

are

contained in $\{\lambda;a_{0}\leq\Re\lambda\leq a_{1}, |^{\alpha}- s\lambda|\leq b_{0}\}$.

Now let

us

consider

(Eq) $P(_{Z}, \partial)u(z)=f(z)$.

We have results concerning the growth properties of solutions of (Eq).

Theorem 4. Suppose that$P(z, \partial)$

satisfies

Condition 1. Let $u(z)\in$

$\mathcal{O}_{(\gamma^{*})(\Omega())}\theta$ be

a

solution

of

(Eq). Suppose that $f(z)\in O_{reg,c}(\Omega(\theta))$.

Thenthere is

a

polydisk$U$ centered at$z=0$ such that $u(z)\in \mathcal{O}_{re_{\mathit{9}^{c’}}},(U(\theta))$

for

any $c’< \min\{c-e_{*}, a_{0}\}$.

Theorem 5. Suppose that $P(z, \partial)$

satisfies

Condition $\mathit{0}$

.

Let $u(z)\in$

$O_{(\gamma^{*})}(\Omega(\theta))$ be a solution

of

(Eq). Suppose that $f(z)\in \mathcal{O}_{reg,c}(\Omega(\theta))$.

Then there is

a

polydisk $U$ centered at $z=0$ and a constant $c$” such

that $u(z)\in \mathcal{O}_{reg,\mathrm{c}^{l\prime}}(U(\theta))$.

We show Theorem 4 by constructing a parametrix and Theorem 5

follows from Theorem 4. The proof theorems and the details of this

paper will be appeared in the forthcoming paper.

We give

some

examples satisfying Condition 1:

(a). Operators of normal type with respect to $\partial_{0}$,

$\partial_{0}^{k^{*}}+\alpha 0<k\sum*a(\alpha z)\partial^{\alpha}$.

(b). Operators of Fuchsian type.

(c). Other concrete examples

are

$I_{d}+z_{00}^{2}\partial+z_{0}\partial^{2}1$

’ $z0\partial^{2}0^{+(_{Z)\partial_{0}}}a+\partial 13$. The present paper follows $\overline{\mathrm{O}}$

uchi [4]. The class of operators

consid-ered in [4] was more strict than that of this paper. The main Theorem

in [4] was the following:

If

$u(z)$ grows at most some exponential order near $z_{0}=0_{\mathrm{Z}}$ that $is_{f}$

for

any $\epsilon>0$ $|u(z)|\leq C_{\epsilon}\exp(\epsilon|z0|-\gamma^{*})$ near _{$z_{0}=0$}, and

if

$f(z)$

hehaves asymptotically $f(z) \sim\sum_{n=0}^{+}\infty f_{n}(Z’)z_{0}n$ as $z_{0}arrow \mathit{0}$ in a sectorial

region $\Omega(\theta)$, where $|f_{n}(z’)|\leq AB^{n}\Gamma(n/\gamma^{*}+1)$, then $u(z)$ has also the

asymptotic expansion like $f(z)a\mathit{8}Z_{0}$ tends to $\mathit{0}$.

It

was

an extension of the main result of [1] and [2]. But in the

present paper we treat a wider class of operators which contains that

of [4]. So even if $f(z)$ has

a

Gevrey type asymptotic expansion, $u(z)$

does not always have. Hence, Theorem 4 in this paper is somewhat

different. Roughly speaking,

if

$u(z)$ grows at most some exponential order near $\{z_{0}=0\}$, and

if

$f(z)$ has the $sl_{\mathit{0}}wl,y$ increasing singularities on _{$\{z_{0}=0\}_{f}$} then the

growth order

_of

singularities

_of

$u(z)$ are also slowly increasing.

We

can

show the results in [4], by using Theorem 4.

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uchi,S., An integral representation _ofsingular solutions and removable

sin-gularities tolinearpartial

_differential

equations, Publ. RIMS KyotoUniv. 26,

735-783 (1990).

[2] $\overline{\mathrm{O}}$uchi, S., The behaviour

of

solutions with singularities on a characteristic

surface

to linear partial

_differential

equations in the complex domains, Publ.

RIMS Kyoto Univ. 29, 63-120 (1993).

[3] $\overline{\mathrm{O}}$

uchi,S., Genuine solution8 and

_formal

solution8 with Gevrey type estimates

ofnonlinearpartial_defferntial equations, J. Math. Sci. Univ. Tokyo 2 (1995),

375-417.

[4] $\overline{\mathrm{O}}$uchi,

S., Singular solutions with asymptotic $ex.panSi_{\mathit{0}}n$

of

linear partial

dif-ferential equations in the complex domain

Sunao $\overline{\mathrm{o}}_{\mathrm{U}\mathrm{c}}\mathrm{H}\mathrm{I}$

Department of Mathematics, Sophia University

Kioicho Chiyoda-ku, Tokyo 102,

JAPAN